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In this paper we prove the existence of the quadratic covariation [f( X), X], where f is a locally square integrable function and Xt = ∫ t 0 us is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. [ABSTRACT FROM AUTHOR]s is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus. [ABSTRACT FROM AUTHOR]

In this paper we show that the local time of the Brownian motion belongs to the Sobolev space $$\mathbb{D}^{\alpha {\text{,}}p}$$ for any p≥2 and 0<α<1/ p. In order to prove this result we first discuss the smoothness and integrability properties of the composition of the Dirac function δ with a Wiener integral W(h), and we show that this composition belongs to $$\mathbb{D}^{ - \alpha {\text{,}}p}$$ , for any α>0 and p>1 such that α+1/ p>1. [ABSTRACT FROM AUTHOR]